Drell-Yan lepton pairs are produced in the process pp → γ^{*}/Z + X, with the subsequent decay of the γ^{*}/Z into lepton pairs. The angular distribution of decay electrons provides information on the electroweak-mixing parameter sin^{2}θ_{W} via its observable effective-leptonic sin^{2}θ_{W} or sin^{2}θ_{eff}^{lept}. The measurement uses 9.4 fb^{-1} of CDF Run II data in the e^{+}e^{−} channel where the pairs are in the mass range above 50 GeV. The value of sin^{2}θ_{eff}^{lept} is found to be 0.23248 ± 0.00053. When combined with the previous CDF measurement using μ^{+}μ^{−} pairs, the result is 0.23221 ± 0.00046.
The Drell-Yan process at tree level consists of two parton level diagrams: qq → γ^{*} and Z → e^{+}e^{−}. Fermions (f) couple to the virtual photon via vector coupling, Q_{f}γ_{μ}, where Q_{f} is the fermion charge (in units of e). The fermion coupling to a Z boson has both vector (V) and axial (A) couplings: g_{V}^{f}γ_{μ} + g_{A}^{f}γ_{μ}γ_{5}. The Born couplings are:
The angular distribution of the leptons is analyzed in the Collins-Soper (CS) rest frame of the ee-pair. The polar angle of the e^{−} is denoted as θ, and the azimuthal angle as φ. In the CS frame, the placement of the z axis is approximately along the direction of the incoming quark. When the ee-pair transverse momentum, P_{T} is zero, the CS and lab coordinates are the same. The angular distribution integrated over φ is
The A_{4} coefficient is parity violating, non-zero at P_{T}=0, and induces an asymmetry in the cosθ distribution. It is a consequence of vector and axial current interference, and there are two sources: γ^{*}-Z interference and the Z boson vector and axial-vector interference. The γ^{*}-Z interference depends on g_{A} (T_{3}). The Z boson vector and axial-vector interference includes g_{V}, so at tree level, the A_{4} contribution from it is proportional to g_{V}^{l}g_{V}^{q}
In this analysis, the forward-backward cosθ asymmetry of the electron pairs is measured as a function of their mass. The asymmetry is defined as:
This plot on the left illustrates the typical behavior of A_{fb} as a function of the lepton-pair mass M. The vertical line is at M=M_{Z}. The label u+d denotes the overall asymmetry. The labels u and d denote the contributions to the overall asymmetry from charge +2/3 and −1/3 quarks, respectively. The denominator for all cases is the total cross section from quarks of all charges. The intercepts at M=M_{Z} are related to sin^{2}θ_{W}. The rise and fall away from M=M_{Z} is from γ*-Z interference and is influenced by the PDFs. These dependencies are illustrated here. |
The Data
The electron candidates used in this analysis can be in
either the CDF central (C) or the forward end plug (P)
calorimeters. The central and plug calorimeters
cover the range, |η_{det}|<1.1 and
1.1<|η_{det}|<3.5 respectively. Each
electron is required to have an associated track and pass
standard CDF electron selection and fiducial cuts. Three
electron-pair topology categories are used with the following
kinematic selections.
The Simulation
PYTHIA 6.2 simulates the tree level process
qq →
γ^{*}/Z → ee with final state QED
radiation (QED FSR), followed by parton showering. This is
followed by CDF II event and detector simulations. The simulated
events are selected, reconstructed, as analyzed as the
data. The generator-level
P_{T} distribution of the γ^{*}/Z
boson is adjusted so that the simulated P_{T}
distribution is the same as in the data. This is done
in two rapidity bins. In addition, the
mass distribution at the generator level is adjusted with
a calculated K-factor: ResBos (CTEQ6.6) to Pythia (CTEQ5L).
Corrections to the Data and Simulation
The same event selection criteria are applied to both
the data and the simulated data. Corrections are applied to
both the data and simulated data for the measurement of
A_{fb}. Event rate corrections are applied to the
simulation to normalize its rates with respect to the data.
In particular, the
pp
collision vertex distributions that affect energy calibrations
are corrected. The distribution of the number of vertices per
event and the location of the collision along the beamline
are adjusted so that they agree with the observed distributions.
The energy scale of the data and simulation are calibrated with an adaptation of the technique used to calibrate the muon momentum in the previous measurement of A_{fb} [PRD 89,072005 (2014), and A. Bodek, Eur. Phys. J. C 67, 321 (2010)]. With this technique, the Z-boson pole mass location in the ee-pair mass distributions of the data and simulation are aligned with the equivalent generator level quantity. The generator level electrons are after QED FSR, are clustered with nearby photons, and their corresponding reconstruction level electrons pass all selection requirements. As there are 1440 calorimeter towers with time-dependent gains, the calibration is done in a series of iterative steps. Time and position dependence of the gains of the data and simulation are removed with the calibrations. The calibrations for the simulation also include the tuning of the calorimeter resolutions so that they agree with those observed in the data.
Because of the track requirement on each electron, the backgrounds are very small in the Z-peak region and do not affect the energy calibrations. However, they are larger at low and high masses. The QCD background shape and level for each of the three ee-pair topologies are derived from the ee-pair mass distributions of the data. EWK backgrounds from WW, WZ, ttbar, W+jets, and Z → ττ are from Monte Carlo (MC) samples which are simulated and reconstructed as data. The difference between the data and the sum of the simulated signal and EWK backgrounds is taken as the QCD background. The QCD background is extracted from a fit over the mass region, 42-400 GeV. The QCD background sample used for background subtractions is derived from the data set used to select ee-pairs for the measurement. However, some of the selections are reversed to obtain a jet or hadron like sample of pairs. As the reverse selection biases the mass distribution, it is reweighted to match the background shape and level obtained in the fit to the data. The results of the background fits are shown below.CC opposite-charge pair mass distribution. The data are the crosses and the red histogram the sum of the simulation and all backgrounds. The backgrounds are: QCD (magenta), Z → ττ (green), W+jets (blue), WW+WZ+ZZ (cyan), and tt (purple). The χ^{2} between the data and sum of the simulation and backgrounds is 56 for 50 bins. |
CP pair mass distribution. The data are the crosses and the red histogram the sum of the simulation and all backgrounds. The backgrounds are: QCD (magenta), Z → ττ (green), W+jets (blue), WW+WZ+ZZ (cyan), and tt (purple). The χ^{2} between the data and sum of the simulation and backgrounds is 50 for 50 bins. |
The following show the results of the energy calibrations and
background fits for the CC and CP topology ee-pairs.
CC opposite-charge pair mass distribution. The crosses are the background subtracted data, and the histogram the simulated Z's. The χ^{2} between the data and sum of the simulation and backgrounds is 214 for 200 bins. |
CP pair mass distribution. The crosses are the background subtracted data, and the histogram the simulated Z's. The χ^{2} between the data and sum of the simulation and backgrounds is 235 for 200 bins. |
CC same-charge pair mass distribution. The crosses are the background subtracted data, and the histogram is the simulated γ*/Z's with charge misidentification. Charge misidentification is reproduced well by the simulation, and so charge misidentification for the CP events is also expected to be properly accounted by the simulation. |
E_{T} distribution for the CC topology electron with the larger E_{T}. The crosses are the background subtracted data, and the histogram the simulated Z's. |
E_{T} distribution for the CP topology electron with the larger E_{T}. The crosses are the background subtracted data, and the histogram the simulated Z's. |
The A_{fb} Measurement
The traditional method to measure A_{fb} utilizies corrected cross sections, σ = N/(LεA), where L is the integrated luminosity, ε the reconstruction efficiency, and A the acceptance. The fully corrected forward-backward asymmetry is given by
A_{fb} slowly increases as |y| increases beyond |y|=1. Consequently, as the angular weighting method requires events for account for this, the measurement must be restricted to a region where there is sufficent acceptance. Thus, the boson rapidity is restricted to |y| < 1.7. This also limits the size of secondary acceptance corrections. The curve labeled PYTHIA shows the (arbitrarily normalized) shape of the underlying rapidity distribution from PYTHIA. |
The raw A_{fb} measurement is shown in the plot on the left, along with the PYTHIA prediction (before QED FSR). The underflow bin is 50-64 GeV/c^{2} and the overflow bin is 150-350 GeV/c^{2}. Only event weighting acceptance corrections are applied. |
The event weighted sums N^{±}_{n} and N^{±}_{d} require resolution unfolding in the electron-pair mass and cosθ. The simulation is used to do the unfolding. Bin-to-bin event migrations are tracked by the smearing transfer matrix n_{gr}, which tabulates the number of selected events produced in mass bin g that are reconstructed in another mass bin r. The following estimator is used as the resolution unfolding matrix: n_{gr}/N_{r}, where N_{r} is the total number of events reconstructed in mass bin r. Because of the different energy resolutions of CC and CP electron pairs, separate transfer matrices are maintained for each topology. The transer matrices also used to estimate the A_{fb} measurement covariance matrix. The separate CC and CP unfolding is combined for the covariance matrix. This method of unfolding requires that the simulated data cosθ distribution match the data. The simulated data cosθ distribution is tuned to match the data. Only adjustments symmetric in cosθ are applied, and the intrinsic asymmetry is unchanged. The default φ distribution is adequately simulated.
The final step of the A_{fb} measurement after the
resolution unfolding, is the correction of the small biases
after event-weighting and unfolding. They are primarily due to
regions of limited kinematic acceptance and detector non-uniformities.
The simulation is used to predict the bias, defined as
the true value minus the estimate of A_{fb} from
the event-weighting method after resolution unfolding.
The true value is the PYTHIA
calculation of A_{fb} with the boson rapidity
kinematic restriction of |y| < 1.7. The estimate is the
A_{fb} obtained with the simulated data. All corrections
applied to the data are applied to the simulated data.
The black crossses are the correction for uncompensated biases after event weighting and resolution unfolding. The significant bias corrections to A_{fb} are under 8% in magnitude and most are 3% or less. The estimated A_{fb} is the event-weighted simulated data after unsmearing. |
The fully corrected A_{fb} measurement is obtained by adding
the bias correction to the event weighted and unsmeared A_{fb}.
Fully corrected A_{fb}. The first bin, denoted as the 'underflow' bin, covers the mass range 50-64 GeV. The last bin, denoted as the 'overflow' bin covers the mass range 150-350 GeV. The uncertainties shown are bin-by-bin unfolding estimate, and not the diagonal elements of the measurement covariance matrix. The vertical line is the mass of the Z-boson. The PYTHIA prediction is with sin^{2}θ_{eff}^{lept} = 0.232. The Powheg-Box prediction, described in the next section, uses sin^{2}θ_{W} = 0.2243 (sin^{2}θ_{eff}^{lept} = 0.2325). |
QCD Calculations:
a) Electroweak radiative corrections
QCD, QED, and weak radiative corrections can be organized to
be separately gauge invariant, and corrections for each can
be applied separately and independently. QED radiative
corrections (with real photons) are not included in the calculation of
A_{fb}. They are included in the simulation physics model,
so QED radiative effects are removed from the measurement of
A_{fb}.
Weak radiative corrections are based on ZFITTER 6.43's e^{+}e^{−} → Z → qq amplitude form factors. The ZFITTER form factors are finite and gauge invariant. Thus photon corrections that involve massive gauge bosons are included in the Z amplitude form factors for gauge invariance. Internal QED radiative corrections to the photon propagator are only from fermion loops, and this correction is another form factor. ZFITTER uses the on-shell renormalization scheme for its form factors. All particle masses are on shell, and sin^{2}θ_{W} = 1 − M_{W}^{2}/M_{Z}^{2} to all orders of perturbation theory. The values of the form factors depend on the input sin^{2}θ_{W} and other standard model parameters.
b) Drell-Yan QCD calculations
The ZFITTER complex-valued form factors are incorporated into QCD
calculations for an enhanced Born approximation (EBA) to the
electroweak couplings. For the form factors, √s is assigned
the mass of the lepton pair. This done for both LO and NLO QCD calculations
of the Drell-Yan process. Operationally, the QCD related portion of
matrix elements are unchanged, and only the electroweak coupling
portions need to be appropriately modified. Two NLO calculations
are used in the calculations of A_{fb}:
Powheg-Box and ResBos. The Powheg-Box calculation is interfaced with
Pythia's parton showering algorithm, and for brevity, the Powheg-Box plus
Pythia combination is denoted as Powheg-Box.
A simple QCD LO, or tree calculation of the Drell-Yan process is used
as a baseline reference for the higher order QCD calculations. For the
Powheg-Box and LO calculations, the NNPDF-3.0 NNLO PDF ensemble is
used (α_{s} = 0.118). The ResBos calculation uses CTEQ6.6.
The EBA form factors modifies the Born g_{A} and g_{V}
couplings:
The Powheg-Box NLO predictions of A_{fb} for various input values of sin^{2}θ_{W} are chosen as the default for comparisons with the measurement. The ResBos and LO calculations are for comparisons and cross checks.
The NNPDF-3.0 PDFs are an ensemble of a 100 probability-based PDFs. They are random samples from the probabilty density distribution of the PDF parameters constrained by measurements. Each PDF of the ensemble is equally probable. Thus, the prediction for an observable is the simple average of the values calculated over the ensemble, and the uncertainty of the observable from PDFs is the rms about the average. If a new measurement is consistent with those used to constrain the PDF parameters, it can be incorporated into the ensemble without regeneration. This is accomplished by re-weighting the ensemble PDFs, numbered 1 to N, with the probabilities of likelihood between the new measurement and the calculations:
Extraction of sin^{2}θ_{eff}^{lept}
The fully corrected A_{fb} measurement is compared with predictions using the χ^{2} statistical measure derived from the measurement error matrix. The error matrix is regulated with a cutoff function. The SVD (singular value decompositon) expansion of the covariance or error matrix in terms of the eigenvalues and eigenvector projection operators of the covariance matrix has eigenvalues that are orders of magnitude smaller than the smallest uncertainty of the measurement. This produces instabilities in the χ^{2} and the cutoff function attenuates those instabilities.
For the EBA-based QCD calculations, the parameter that specifies the electroweak mixing is sin^{2}θ_{W}. As the measurement is directly sensitive to sin^{2}θ_{eff}^{lept}, the best-fit value of sin^{2}θ_{W} only represents the corresponding sin^{2}θ_{eff}^{lept}, which is independent of the standard model parameters used for the EBA form factors. However, the interpretation of the best-fit value of sin^{2}θ_{W} only has meaning within the context of the model assumptions used.
The χ^{2} is calculated for a series of A_{fb} calculations, denoted as scan templates. The series of scan points are fit to a generic χ^{2} functional form
An example of the χ^{2} scan over the Powheg-Box NLO templates. The inverted triangles are the scan points, and the solid curve is the fit to those scan points. |
Template (Measurement) |
sin^{2}θ_{eff}^{lept} | sin^{2}θ_{W} | χ^{2} | |
Powheg-Box NLO, NNPDF-3.0 | 0.23248±0.00049 | 0.22428±0.00048 | ±0.00018 | 15.4 (15) |
ResBos NLO, CTEQ6.6 | 0.23249±0.00049 | 0.22429±0.00047 | 21.3 (15) | |
Tree LO, NNPDF-3.0 | 0.23250±0.00049 | 0.22430±0.00047 | ±0.00021 | 21.5 (15) |
Pythia, CTEQ5L | 0.23207±0.00046 | 24.6 (15) | ||
(CDF 9.2 fb^{−1} A_{fb}^{(μμ)}) | 0.2315±0.00010 | 0.2233±0.00009 | 21.1 (16) | |
(CDF 2.1 fb^{−1} A_{4}^{(ee)}) | 0.2328±0.0011 | 0.2246±0.0011 | ||
(LEP-1 and SLD A_{FB}^{0,b}) | 0.23221±0.00029 | |||
(SLD A_{l}) | 0.23098±0.00026 |
While the sin^{2}θ_{W} values from Powheg-Box and ResBos calculations are the same, the χ^{2} values differ. The A_{fb} difference relative to the Pythia CTEQ5L calculation (sin^{2}θ_{eff}^{lept} = 0.232), ΔA_{fb} = A_{fb} − A_{fb} (Pythia), shows differences in the best-fit A_{fb}.
ΔA_{fb} for the measurement, the Powheg-Box A_{fb} with the default PDF of NNPDF-3.0, and the ResBos A_{fb} with CTEQ6.6. The ΔA_{fb}=0 line is Pythia with CTEQ5L. The uncertainties are the bin-by-bin unfolding estimates, which are correlated in the Z-pole region. However, this illustrates the dependence of the A_{fb} measurement to PDFs. |
The Powheg-Box template fit parameters of χ^{2} vs sin^{2}θ_{W} for all of the ensemble PDFs of NNPDF-3.0. The A_{fb} measurement and templates span 15 mass bins. This illustrates that the measurement is consistent with the other input measurements of NNPDF-3.0. |
Uncertainities of the Extracted sin^{2}θ_{eff}^{lept}
The uncertainties to sin^{2}θ_{eff}^{lept} are from the measurement of A_{fb} and the template predictions of A_{fb}. For the measurement, these sources are considered: the electron energy scale plus resolution, and the backgrounds. For the predictions, two source are considered: QCD scales and NNPDF-3.0 PDFs. For the inference of sin^{2}θ_{W} based on the measurement, the uncertainty of the top mass input to ZFITTER is considered.
The relative difference between simulation and data energy scales allowed by the data is taken as the energy scale uncertainty. The energy scale and resolution of central and plug electrons are well constrained by the precision of the Z-peak within the electron-pair mass distribution. The uncertainties of the energy scale and background fits are propagated to the extracted value of sin^{2}θ_{W}. The effect from limitations of the energy resolution model in the simulation is estimated to be negligible.
The QCD scale undertainty is taken to be difference between the Powheg-Box NLO and Tree (LO) values of sin^{2}θ_{W}.The value of the inferred sin^{2}θ_{W} based on the measurement depends on the SM input parameters of ZFITTER. The influence of the top mass measurement uncertainty, 173.2±0.9 GeV/c^{2}, is included as a systematic uncertainty. This uncertainty is denoted as the form factor uncertainty.
The uncertainties are summarized in the table below.
In this table, 'Data' denotes the uncertainties from the
A_{fb} measurement, and 'Pred' denotes the
prediction uncertainties.
Source | sin^{2}θ_{eff}^{lept} | sin^{2}θ_{W} |
Data: Measurement | ±0.00049 (stat) | ±0.00048 (stat) |
Data: Energy scale | ±0.00003 (syst) | ±0.00003 (syst) |
Data: Backgrounds | ±0.00002 (syst) | ±0.00002 (syst) |
Pred: QCD scales | ±0.00002 (syst) | ±0.00002 (syst) |
Pred: QCD PDFs | ±0.00019 (syst) | ±0.00018 (syst) |
Pred: Form factor | − | ±0.00008 (syst) |
Results
The results of the extraction are summarized below.
The interpretation of sin^{2}θ_{W} and M_{W} only has meaning within the context of the standard model parameters used for the EBA form factors. The context is similar to the environment for standard model fits of Z-pole measurements by LEP-1/SLD [Phys. Rep. 427, 257 (2006)], but with the exception that the Higgs boson mass parameter is set to 125 GeV/c^{2}.
The A_{fb} measurement using ee-pairs and the previous CDF measurement using μμ-pairs PRD 89, 072005 are over different kinematic ranges: |y_{ee}| < 1.7 and |y_{μμ}| < 1. For the combined result on the electroweak-mixing parameter sin^{2}θ_{eff}^{lept} (sin^{2}θ_{W}), A_{fb} templates are calculated separately, and the joint χ^{2} of the individual comparisons used to extract the combined electroweak-mixing parameters.
The results of the combination are summarized in the table below.
The methods of evaluation for the
χ^{2} between the A_{fb} measurements and its
templates are unchanged. The Powheg-Box NLO and Tree calculations
utilize the NNPDF-3.0 ensemble of PDFs.
Template | sin^{2}θ_{eff}^{lept} | sin^{2}θ_{W} | χ^{2} | |
Powheg-Box NLO | 0.23221±0.00043 | 0.22400±0.00041 | ±0.00016 | 35.9 (31) |
Tree LO | 0.23215±0.00043 | 0.22393±0.00041 | ±0.00016 | 37.4 (31) |
The Powheg-Box fit parameters of χ^{2} vs sin^{2}θ_{W} for all of the ensemble PDFs of NNPDF-3.0. This shows that the combined measurement is consistent with the other input measurements of NNPDF-3.0. |
The uncertainties for the combination are summarized in the table below.
In this table, 'Data' denotes the uncertainties from the
A_{fb} measurement, and 'Pred' denotes the
prediction uncertainties.
Source | sin^{2}θ_{eff}^{lept} | sin^{2}θ_{W} |
Data: Measurement | ±0.00043 (stat) | ±0.00041 (stat) |
Data: Energy scale | ±0.00002 (syst) | ±0.00002 (syst) |
Data: Backgrounds | ±0.00003 (syst) | ±0.00003 (syst) |
Pred: QCD scales | ±0.00006 (syst) | ±0.00007 (syst) |
Pred: QCD PDFs | ±0.00016 (syst) | ±0.00016 (syst) |
Pred: Form factor | − | ±0.00008 (syst) |
The results of the combination are summarized below.
The corresponding sin^{2}θ_{eff}^{lept} measurements from LEP-1/SLD [Phys. Rep. 427, 257 (2006)], and from the the A_{fb} measurements of D0 [PRL 115, 041801 (2015)] and CDF [PRD 89,072005 (2014)] using their full Run II datasets are:
Comparison of sin^{2}θ_{eff}^{lept} that includes latest LHC results from CMS [ Phys. Rev. D84 112002, 2011 ], ATLAS [ J. High Energy Phys. 09 (2015) 049 ], and LHCb [ J. High Energy Phys. 11 (2015) 190 ]. The LEP-1+SLD Z-pole entry is the combination of their six Z-pole measurements. |
The inferred value of sin^{2}θ_{W} is usually expressed as an indirect W-boson mass value. There are other indirect W-boson mass results, including those from LEP-1 and SLD which are from standard model fits to Z-pole measurements with the top quark mass, and there are direct W-mass measurements from the Tevatron and LEP-2 [Phys. Rev. D86, 010001 (2012): PDG W mass review, 2015 update].
All measurements except for 'TeV and LEP-2' are indirect W-mass measurements that use the standard model (on-shell scheme). NuTeV is the neutrino neutral current measurement [ PRL 88, 091802 (2002); PRL 90, 239902(E) (2003) ] from the Tevatron. |